Bond Pricing With and Without Default

Bond Pricing with/without Default

Important language difference between loans and bonds:

• Loan: Borrower (Buyer) receives $X$ and pays installments of $A$ to the lender (issuer, seller),
• Bond: Borrower (Issuer, seller) receives $X$ and pays installments of $C$ and a final payment of $C+X$ to the lender (buyer)

How a Bond Works

At time $0$ the lender gives the borrower a “principal” amount of \X$.

At time periods $1$ through $N-1$ the borrower repays equal coupons \C$.

At the final repayment period $N$ the lender repays not just a coupon of \C$butalsotheinitialprincipal$$X$.

Government Bonds

When governments of first-world countries (Canada, US, Germany, UK, etc) borrow money, we assume that they are certain to repay their debts fully. That is because they tend to borrow in heir own currency, the printing of which they control.

Risk-free interest rate, is the rate of interest a lender would be willing to accept for lending money to a risk-free borrower.

If the borrower is sue to repay the loan, and the interest rate if $r\%$ per period, then the coupon \C$hastobe$$rX$.

Here we assume that the risk-free interest rate is known, fixed over time.

Corporate Bonds

When corporations borrow money, there is always a risk that they will default on their debt. That is, they may not be able to make all their coupon payments, or repay the principal at maturity.

Thus, corporate bonds have to pay a higher coupon than government bonds to compensate lenders for the risk of default.

Setting up Our Problem

Assumption 3

• The event of default in each period is independent from the other periods (note, once a company defaults, it remains in default).
• Borrowers only default the instant before their next payment is due
• In the event of default, lenders receive a fraction of which is owed \R(C+X)$,where$R$ is a known constant (known as the recovery rate)

Context:

• Defaulted bonds are sold to Culture funds, which usually pay recovery rates of about 40%. The culture funds fight the court battles.

Determining the Probability of Default

We’re going to assume, the probability of default and the recovery rate is known.

Using Difference Equations to Find the Coupon

• Let $V_0$ be the expected value of the bonds immediately after issuance (time $0$)
• Let $V_k$ be the expected value of the bond immediately after the $k$-th coupon payment is made, $k=1,\dots ,N-1$. That is, there are $n-k$ remaining payments.
• Note that $V_k$ contains all the information about possible defaults at times $k+1,k+2,\dots ,N$
• Assume the principal is repaid a bit later than the final coupon, so that $V_N$ is the value of the principal, that is, let $V_N=X$.

Proposition 9

Let $R$ be a fixed number in $[0,1]$ and $p$ the probability of default. The correct coupon value is:

$$C=X\frac{r+p(1-R)}{1-p(1-R)}$$

Proof of Proposition 9

First, we need to find the value of the bond at issue, $V_0$

In order to compute $V_k$, all we need to consider is the possibility of a default at time $k+1$ as the effect of all other default events, after that, are lumped into the value of $V_{k+1}$. A default at time $k+1$ occurs with probability $p$, returning $R(C-X)$ at time $k+1$.

If there is no default (with probability $1-p$), a coupon of $C$ is received at time $k+1$ and the remaining payments, adjusted for the possibility of future default, are worht $V_{k+1}$.

Note that the possibility of default does not affect the value of $V_N$ since it is not possible to have a default at time $N+1$.

Since, all of those payments are at time $k+1$, we need to bring them back to time $k$ by dividing them by $1+r$ to account fo the time value of money.

Since the lender computes the expected value of their cashflows, we have a difference equation:

$$V_k=(1-p)\frac{C+V_{k+1}}{1+r}+p\frac{R(C+X)}{1+r}$$

for $k=0,1,2,\dots ,N-1$ with boundary condition $V_N=X$.

First, we non-demensionalize by letting $w_k:=\frac{V_k}{X}$ and $c:=\frac{C}{X}$, where $c$ now has the interpretation of a “coupon rate.”

$$w_k=(1-p)\frac{c+w_{k+1}}{1+r}+p\frac{R(c+1)}{1+r}$$ $$w_N=1$$

We also make the time-reversal transformation, that is, we let $v_{N-k}:=w_k$

$$V_{N-k}=(1-p)\frac{c+V_{N-k-1}}{1+r}+p\frac{R(c+1)}{1+r}$$

and note that the initial condition become $v_0=v_N=1$.

Finally let $j:=N--1$ and note $j=0,\dots ,N-1$

$$v_{j+1}=(1-p)\frac{c+v_j}{1+r}+p\frac{R(c+1)}{1+r}=\frac{c(1-p)(1-R)}{1+r}+\frac{pR}{1+r}+\frac{1-p}{1+r}{v}_j$$

Our goal is to find $v_N$. In order to simplify the notation, define

$$a:=\frac{1-p}{1+r}b:=\frac{c(1-p)(1-R)}{1+r}+\frac{pR}{1+r}$$

so that

$$v_{j+1}=b+a{v}_j$$ $$b=1-a$$ $$v_j=a^j+\frac{1-a^j}{1-a}$$

Proposition 10

Let $R$ be a fixed number in $[0,1]$ representing the recovery rate as per $R\times A$, $p$ the probability of default, $X$ the loan size, and $r$ the short rate. The correct discrete payments $A$ shall be:

$$A=\frac{(r+p{)}^X}{1-p(1-R)}\frac{1}{1-\frac{(1-p{)}^N}{(1+r{)}^N}}$$

Proof of Proposition 10

Similar to previous results, solve the following

$$V_k=\frac{1-p}{1+r}(C+V_{k+1})+\frac{p}{1+r}R\times C$$

Subject to

$$V_N=0$$

TODO